A SELECTED BIBLIOGRAPHY OF MATERIALS ON GEOMETRY USEFUL TO HIGH SCHOOL TEACHERS AND STUDENTS OF MATHEMATICS By GARELD DEAN WHITE Bachelor of Science Northwestern State College Alva, Oklahoma 1954 Submitted to the faculty of the Graduate School of the Oklahoma State University in partial fulfillment of the requirements for the degree of ¥JASTER OF SCIENCE August, 1960 A SELECTED BIBLIOGRAPHY OF M.l\.TERIALS ON GEOMETRY USEFUL TO HIGH SCHOOL TEACHERS AND STUDENTS OF MATHEMATICS Report Approved: Dean of the Graduate School ii PREFACE This report is primarily intended to aid teachers or prospective teachers of elementary or secondary school mathematics in the selection of appropriate geometrical material to be used as supplementary exercises for the student or as self-instructional material for themselves. The author is indebted to his adviser, Dro James H. Zant, for his helpful assistance and guidanceo iii TABLE OF CONTENTS Chapter Page 0 .. 0 0 0 0 0 Cl 0 0 C O O 4 Q • Q I. INTRODUCTION., o " 0 " • 0 1 II. BIBLIOGRAPHY 0 c:i • e o 0 0 "ll Q 0 0 0 Q 0 0 0 .. .. 0 " " .5 1. History • 0 0 0 0 • 0 .. 0 0 0 0 Q .. 0 0 0 0 0 .. 0 0 5 Plane and Solid Euclidean Geometry., 0 0 0 0 6 2. • .. " " Plane and Solid Analytic Geometry 0 0 0 0 0 0 ., 12 3. • " Modern and Non-Euclidean Geometry 0 0 0 0 0 4. .. " • .. 13 Advanced Geometry e 0 0 0 5. • • • • • • • • • . " • . 1.5 6. Enrichment and Professional Material., • 0 0 0 0 ct .. 17 III. SUMM..ARY., tlflOQl'OOOOOO&O QO OOOQQO 0 " • 0 0 0 19 iv CHAP~RI INTRODUCTION This bibliography is for the specific purpose of acquainting mathematics teachers with various sources of reference material on the subject of geometry. It not only contains a source of a partic­ ular work, but it also includes a short description of the work itself and other information which might be useful to the teacher. Three principal methods were used in determining what books to include in this worko A preliminary list was obtained by asking several leading mathematics educators from various colleges and uni­ versities throughout the United States to suggest books and other materials that should be included in an undertaking of this nature. A second list was obtained from suggested mathematical materials ob­ tained from various educational publications. A final list was selected by the author. This bibliography was then prepared from these three listings. Not all the materials found in the above three lists are in­ cluded in this work. After a review of each article or book was taken, those which seemed most useful in enabling the teacher to teach and the student to learn fundamental mathema.tical concepts were selected to be included in this reporto Several of the books listed here are strictly for the professional. The need for such a collection of data as is compiled in this report is justified by the ever-increasing emphasis that has been placed in recent 1 2 yeRrs and is continuing to be placed on the teaching of the right kind of mathematics in the upper elementary and secondary schools of our country. Much of the material included in this work is rather recent and contains to a large extent articles concerned with the modern ap­ proach to mathematicso This is not to imply that our traditional courses are no longer valuable but to point out that they need to be at least supplemented and in many cases revised if they are to meet the ever­ changing needs of our society. It follows that in a revised program of this sort, a very important asset to any teacher would be not only an adequate knowledge of the new approach itself, but also a wide source of reference material on the subject. A source of material is included here; the other asset is not included but it may be acquired through proper self-instruction. It was this idea of self-instruction and individual effort that Professor G. H. Hardy stressed in 1925 when i11 the Presidential Address to the Mathematical Association he said, 11 In mathematics there is one thing only of primary importance, that a teacher should make an honest attempt to understand the subject he teaches as well as he can, and expound the truth to his pupils to the limits of their patience and capacity, 11 The need for a bibliography of this type can be further justified by the inadequate preparation of many of the instructors teaching math­ ematics in our schools today. Many are teaching subjects for which they were not prepared to teach. Others, who were supposedly prepared to teach mathematics, find themselves not yet ready to do a good job due to the fact that the courses they received in college were not sufficient or of the proper type for present-day ideas. This might have been 3 brought about by many schools leaving the training of mathematics teachers too much to the control of the education departments where emphasis was placed on methods rather than on the fundamental concepts and structure of mathematicso Perhaps this was what Professor G. H. Hardy had reference to when, in the same Presidential Address as mentioned above, he set forth the following ideas. "It is obvious that a great part of what is taught in schools and universities under the title of geometry is not geometry, or at any rate mathematical geometry, at all, but physics or perhaps philosophy." "There is not, a.nd cannot be, any question of a mathematician prov­ ing anything about the physical world; there is one way only in which we can possibly discerm its structure, that is to say, the laboratory method, the method of direct observation of the facts. tt ttSchool geometry is not a well-defined subject, a rational exposi­ tion of a particular geometrical system, but a collection of miscella­ neous scraps, a selection of airs from different pieces, strung together in the manner which experience shows to be the most enlivening ••• o••••" 11 I think that it is time that teachers of geometry became a little more ambitious. It seems to me regrettable that students are not given the opportunity, while still at school, of learning a good deal more about the subject-matter out of which modern geometrical systems are built. It is probably easier and certainly vastly more instructive than a great deal of what they are actually taught." It is evident that the time for better understanding of the funda­ mental concepts of mathematics on the part of both teacher and student has come and it is hoped that this bibliography will be of some value to 4 all people interested in the study of mathematics and in particular to high school teachers and students of geometry. CHAPTER II BIBLIOGRAPHY The references included in this report are divided into specific categories for ease in the selection of a particular work. It is realized that in many cases the textbooks and articles included in this bibliography may be improperly placed, but the exact classifica- tion of each work seemed rather difficult and was considered a minor detail with respect to the over-all purposes of this reporto Even though all of the works listed in this bibliography can offer much toward the improvement of general understanding of geometry, those which seem to offer most in terms of present-day needs of teachers and students of mathematics are indicated by an asterisk. 1., History *Eves, Howard. An Introdµction to the History of Mathematics. New York: Rinehart and Company, Incorporated, 1953, 422 p. This is an excellent book on the history of mathematics and it is especially good for teacher-education purposes. Only a knowledge of elementary mathematics is needed to understand this book which points out in an interesting manner the chrono­ logical growth of mathematics from its primitive form to the possibilities of the modern mathematics superstructureo *Gould, s. H. "Origins and Developments of Concepts of Geometry. 11 National Council of Te?-chers of Mathematics karbook, XXII, 1957, pp. 273-305. As the title indicates, the author gives a fine account of the origin and development of geometry from Euclid to the non­ Euclidean geometry of Gauss, Lobachevski, and Bolyai and the finite but unbounded space concept of Riemanno 5 6 Kline, Morris. Mathematics In Western Culture. New York: Oxford University Press, 1953, 484 p. This book illustrates the cultural force that mathematics has made in Western civilization. Such a fine book as this should be available to all energetic students of mathematics. *Reves, George E. "Outline of the History of Geometry. 11 School Science and Mathematics, III, 1952, pp. 299-309. This article contains a short descriotive outline of the history of the development of geometry fr~m 3100 B.C. to the present. Mention is made of various individuals or groups of people and their contribution to the field of geometry. Smith, David Eugene. History of Mathemati~~. New York: Dover Publica­ tions, Incorporated, 1925. Here is a chronological development of mathematics from number concepts to the calculus. This should be very beneficial at the senior high school levelo 2. Plane and Solid Euclidean Geometry Anderson, Raymond. Romping Thru Mathematics. New York: A. A. Knopf, 1947, 152 p. Many explanatory drawings, diagrams, and charts to make clear various topics on geometry, algebra, and other phases of mathematics are included in this work. *Birkhoff, George David and Ralph Beatley. Basic ~ometry. New York: Chelsea Publishing Company, 1959, 288 p. This is one of the better textbooks in print today concerned with a modern approach to Euclidean geometry.